Math10th grade LEARNING OBJECT LEARNING UNIT Trigonometry, a study of Identifying properties of triangles the measurement of the through function S/K SCO: Understand the theorem. SKILL 1. Identify triangles with sides of equal measurement. SKILL 2. Identify triangles with equal . SKILL 3. Distinguish the metric characteristics of isosceles right triangles. SKILL 4. Understand the hypothesis stated in the isosceles triangle theorem and its inverse. SKILL 5. Solve geometric situations by applying the isosceles theorem. SCO: Identify the 30-60-90 theorem. SKILL 6. Understand the relationship of constant ratio between the and a leg. SKILL 7. Identify the relationships of measurement between the angles and catheti of the triangle. SKILL 8. Illustrate 30-60-90 triangles by applying the relationship between the measurements between the catheti. SCO: Understand the . SKILL 9. Recognize the hypotenuse as the longest leg in a right triangle. SKILL 10. Interpret the Pythagorean Theorem in problem solutions. SKILL 11. Solve measurement situations involving right triangles. SKILL 12. Make an argument supporting the use of the procedure in problem solving. Language English Socio cultural context of 10th grade students will find situations in their the LO context related with the measurement of heights and lengths of sides of physical structures related to construction, such as pyramids, towers, ramps and others. Curricular axis Variational thinking and analytic and algebraic systems. Spatial thinking and geometric systems. Standard competencies  Recognize and contrast properties and geometric relationships used to demonstrate basic theorems. Background Knowledge  Interpretation of situations by means of the Pythagorean Theorem.  Classification and categorization of triangles. Basic Learning  Use the Pythagorean Theorem to verify if a Rights triangle is a right triangle or not, as well for problem solving. English Review topic Simple Present vs Present Perfect Glossary  Triangle: A flat figure formed by three vertexes and three sides.  Straight Triangle: A triangle that has an internal right angle.  Isosceles Triangle: A triangle that is defined by having two sides of equal length.  Catheti: The sides opposing the acute angles of the right triangle.  Hypotenuse: The diagonal, or longest side, of the right triangle.  Pythagorean Theorem: The theorem that states that any right triangle presents the relation 푐2 = 푎2 + 푏2, c being the hypotenuse and a and b being the catheti.  Length: The distance between two points.  Right angle: An angle that measures exactly 90°. Vocabulary Box  Acute: (Geometry) a. (of an angle) less than 90°. b. (of a triangle) containing only acute angles.  Congruent: (Geometry) Coinciding at all points when superimposed.  Foreman: A person in charge of a particular department, group of workers, etc., as in a factory or something similar.  Fulfill: To satisfy (requirements, obligations, etc.)  Oblique: Neither perpendicular nor parallel to a given line or surface; slanting, sloping.  Obtuse Angle: An angle greater than 90° but less than 180° (Terms retrieved from the website www.dictionary.com)

NAME: ______GRADE: ______

Printable Resource

Introduction

Triangles and their importance

Very relevant knowledge has always been linked to triangles. From the study of these geometric shapes many aspects of engineering are derived, such as methods used to calculate distance or center of gravity, navigation systems, tools and and mechanisms for work, among others. Pyramids represent an example that allows us to distinguish these various elements.

Figure. Pyramids.

This image shows different aspects: first, all of the visible sides are isosceles triangles (remember that there is a theorem about their characteristics). The second aspect to a famous Greek mathematician: Thales of Miletus. He applied an ingenuous method to calculate the height of a pyramid with the help of trigonometry and the similarities between triangles, using their shadows and the height of a wooden stick as a means of Figure. Right triangles in a pyramid. comparing them.

Additionally, elements of the Pythagorean Theorem can also be used. Pythagoras was another ancient mathematician and philosopher. The theorem named after him can be applied to right triangles such as the ones showed in the next image. Undoubtedly, this is proof of its importance since early times.

Looking at the following structures, can you see the triangles? Why do you think they are so important?

Triangles are present in the basic principles of geometry because every single polygon can be broken down into or formed by triangles. We will study some of the theorems regarding triangles and their applications.

Objectives  To recognize the properties that satisfy the requirements of some special right triangles to be considered as such.  To identify the properties of isosceles triangles.  To distinguish the 30-60-90 theorem and the relationships between the measurements of each side.  To apply the Pythagorean Theorem for the task of finding measurements.

Very important!

The sum of the internal angles of any triangle is exactly 180°.

Activity One

Isosceles triangles Oscar and Miguel are friends from school and are having a conversation. Miguel: Hello Oscar, how are you? I have not seen you lately, why have you not come to school? Oscar: My parents and I were travelling, and we returned yesterday. What topics have you

Figure. covered? Miguel: We have worked a lot on geometry. Our last class was about triangle theorems. Oscar: Is it a vast topic? Could you help me?

Miguel: Of course, I will explain it to you. We should begin with the isosceles triangle theorem. First, we must know if you have any background knowledge. Answer this question: what is an isosceles triangle? Oscar: Oh, I know that! I will draw it and explain it to you. An isosceles triangle is one with two equal sides, called catheti or legs, like the one I just drew. Miguel: Yes, very good, but you are missing something. Oscar: Let me think… Errrr, I cannot think of anything! Miguel: When we talk about triangles, we say that the angles in the base are equal, or in other words, are congruent. Oscar: Oh yes! That is one of the characteristics.

Miguel: This characteristic defines the isosceles triangle theorem, which is:

“In an isosceles triangle, the angles opposing the equal sides are equal” .

And is written as: ∡퐴퐵퐶 ≅ ∡퐴퐶퐵 Oscar: Ok, that is easy, but what is its function? Miguel: Look at an example: Find the value of the missing angles in the following figure:

Given that the angle ∡퐵퐴퐶 = 50° Oscar: By definition of the theorem, we know that the opposite angle is equal: ∡퐴퐵퐶 = 50°, and the value of the angle ∡퐴퐶퐵 must be 80°, because that is the value we need to fulfill the sum of 180°. Miguel: Very good, now I can complete the homework assignment.

Did you know that?

There is a different way to classify triangles, which is as either acute or obtuse. The first one is characterized by having all their internal angles be less than 90°; and the last one by having one of the angles larger than 90°.

°

Let´s get to work! Matching activity

Up next, you will find the value of the non-congruent angle ∡퐵퐴퐶. Based on the fact that we are talking about an isosceles triangle, find the value of the congruent angles.

Angle ∡푩푨푪 Value of congruent ∡ 50° 37.5° 10° 52.5° 25° 25° 130° 77.5° 75° 85° 105° 65°

“Word blaster” activity Define the following concepts:

 Triangle

 Isosceles

 Sides

 Angles

 Congruent

Fill in the blanks to complete the definition of an isosceles triangle

A ______is ______if it has a pair of ______that are ______, and two of its ______must also fulfill this condition.

Activity 2

Right Triangles and their applications

Miguel: During another class that you missed we covered the 30-60-90 and the Pythagorean Theorems. The first one is a particular case of right rectangles and the second one is a generalization. Oscar: What did you do during this class? Miguel: Teacher Alejandro began with characterization. First, you must distinguish the right triangles from the other.

Figure. Triangles.

Which ones are right triangles? It is possible to find them just by looking at them. Oscar: That is easy; they can be distinguished by the presence of a right angle, meaning 90°. For example, you can see that the purple and red triangles satisfy this requirement.

Miguel: Very good, that is the idea. Now, what is the function of each theorem?

The 30-60-90 theorem is applied to particular triangles, meaning those that possess this special characteristic, and it is used to find the value of each side, according to the following relationship:

“The length of the hypotenuse is twice the length of the shortest cathetus, and

the length of the longest cathetus is √3 times the length of the shortest cathetus”

Look at an example:

We need to find the height of an electric tower. We have the following information: the horizontal distance from the base of the tower to a reference point, as well as the angle formed from the reference point to the top of the tower, as shown in the figure. Oscar: I see. In this case we need to relate the distances, because if we extract the triangle from the figure, we get a 30-60-90 triangle. Then we develop the situation accordingly.

Figure. Electric tower.

Based on the relationship and the data acquired we have the following information: “The length of the longest leg is √3 times the length of the shortest leg”. Therefore we know that the height is the longest leg: 50 ∙ √3 = 86.6 ≈ 87푚. I know that you did not ask this, but the oblique distance, or hypotenuse, is 100m long, according to the definition of the theorem; “…twice the length of the shortest cathetus”

Miguel: Very good! You have learned this quickly.

About the Pythagorean Theorem Miguel: Ok, now let´s cover quickly the Pythagorean Theorem. I will be brief: The Pythagorean relations in the theorem can be summarized in the following three expressions:

Each of these expressions has a specific use. Apply them depending on if you need to find the hypotenuse, or either of the catheti.

On the other hand, the teacher introduced the topic using the following multimedia resource:  https://www.youtube.com/watch?v=m2wTLZfQpWc.

He also allowed us to see a contextual application with the following video:  https://www.youtube.com/watch?v=nA8Yj1cB8iY.

Oscar: All right, very specific. I think everything is clear now, especially with the help of the video. Miguel: Yes! Now, let´s work!

Did you know that?

Because of the relationships established in the Pythagorean Theorem, trigonometry can determine the sine, cosine, and tangent of any right triangle.

Practice Activity Working in groups, solve the following exercises:

1. 30-60-90 Theorem

Suppose that 푎 is the shortest cathetus, 푏 the longest cathetus, and ℎ the hypotenuse. Complete the following chart with the missing information, applying the required theorem (some data is presented as decimals and not as exponentiation powers)

풂 풃 풉 2 4 ∙ √3 24 3 6

1. Pythagorean Theorem

Fill in the blanks with the correct values.

Suppose that 푎 is the shortest leg, 푏 the longest leg, and ℎ the hypotenuse in each of the following situations. Problem situations:

a. The value of the hypotenuse of a triangle whose leg 푎 is 9cm and leg 푏 12cm.

ℎ = _____ cm

b. A skating ramp has the values presented in the following image:

Figure. Ramp. Find the height.

푎 = _____ cm

c. A construction foreman rests a 5 m ladder in the top of the house, and places the base horizontally at a distance of 3 m from the wall to fix a problem on the roof. At what height is the roof found?

Figure. Ladder. 푏= _____ m

Abstract

Homework

Working in groups of a maximum of 3 students each, analyze the following situations and suggest a solution for each one.

Value of Angle ∡푩푨푪 congruent ∡ 56° 15° 75° 89° 72° 5°

1. The height of a soccer goal is 2.5 m and the distance from the goal line to the penalty spot is approximately 10 m. If a player kicks the ball with enough power that it hits the horizontal bar, what distance is the ball travelling?

Figure. Soccer goal.

2. Find the length of the beam of light.

Figure. Beam of light.

 Investigate what Pythagorean triples are and share your explanation with your classmates.

Evaluation

Embedded questions:

Complete each statement based on the different definitions of the theorem of isosceles triangle, 30-60-90, and Pythagoras:

1. In the 30-60-90 theorem, the length of the hypotenuse is twice the length of the ______side. Feedback: In the 30-60-90 theorem, the length of the hypotenuse is twice the length of the shortest side.

2. In an isosceles triangle, the angles opposite to equal sides are _____. Feedback: In an isosceles triangle, the angles opposite to equal sides are equal.

True or false questions.

Choose true (T) or false (F) in each sentence, as appropriate: Take into account the relation that can exist in each statement.

It is correct to say that: 3. It is impossible to construct an isosceles triangle with a right angle. Answer key: False. It is possible. However, it must be an angle not consistent with the other angles. 4. In a right triangle the legs forming the right angle are called sides. Answer Key: True. Sides are adjacent to each other and the angle formed is opposite to the hypotenuse. 5. In a right triangle the hypotenuse will always be the side of greatest length. Answer Key: True. The hinge and triangle inequality theorems determine that the side opposite to the greatest angle is the side of greatest length. 6. There are right triangles with 1 angle greater than 90°. Answer key: False. It is impossible because the sum of its angles is 180°, and the others must be minor because there is already a (90°) straight angle.

Problem zone:

Multiple-choice questions with a single answer: S/K: SKILL 5. Solve geometric situations by applying the isosceles right triangle theorem

7. It is very common to find structures inspired by triangles in engineering, like the one shown below. An engineering assistant has to find the top angle of the structure. For this purpose, he uses a tool to measure angles and is able to determine that one of the angles of the base is 62.5°, for this reason and based on the isosceles triangle theorem we can infer that the top angle is:

a. 45° b. 55° c. 65° d. 75° Answer key: b. Using the isosceles triangle theorem, we have that the angles of the base are equal; then, their sum is equivalent to 125°, and finally we need 55° to complete 180°. So 55° is the measure of the top angle.

S/K: SKILL 11: Solve measurement situations involving right triangles.

8. Pythagoras Theorem has infinite uses in architecture and construction. It is important to consider the following conditions to apply it: first, we must evidence a right triangle; and second, we must know the length of the two sides of the triangle to find the third one. Based on this information, find the height of the statue of Christ presented in the image:

e. 15푚 f. 12푚 g. 10푚 h. 8푚 Answer key: b. If we apply the Pythagorean Theorem we know that 푏2 = ℎ2 − 푎2, then we have 푏2 = 132 − 52, which will result in the value of 푏 = 12푚

S/K: SKILL 10. Interpret the Pythagorean Theorem in problem solutions.

9. A firefighter truck has a ladder that can be raised 8 m vertically from its base, at a distance of 6 m from the buildings it rests upon. A fireman used the Pythagorean theorem ℎ2 = 푎2 + 푏2 to know the total distance the ladder can be extended. Based on the information gathered, the fireman concludes that the maximum distance for the ladder is:

a. 15푚 b. 12푚 c. 10푚 d. 8푚 Answer key: c. If we apply the Pythagorean Theorem we know that ℎ2 = 푎2 + 푏2, then we have ℎ2 = 62 + 82, which will result in the value of ℎ = 10푚

S/K: Understand the hypothesis stated in the isosceles triangle theorem and its inverse

10. An architecture student makes a scaled drawing of a fighter aircraft known as phantom aircraft. The following measures were extracted: 퐴퐶̅̅̅̅ = 9푐푚 and ∡퐶퐴퐵 = 35°. Based on this information, find the triangle properties and infer the value of 퐵퐶̅̅̅̅ and ∡퐴퐶퐵.

a. 퐵퐶̅̅̅̅ = 6푐푚 and ∡퐴퐶퐵 = 35° b. 퐵퐶̅̅̅̅ = 9푐푚 and ∡퐴퐶퐵 = 110° c. 퐵퐶̅̅̅̅ = 6푐푚 and ∡퐴퐶퐵 = 110° d. 퐵퐶̅̅̅̅ = 9푐푚 and ∡퐴퐶퐵 = 35°

Answer key: b. Using the isosceles triangle theorem, the referenced side 퐴퐶̅̅̅̅ is consistent with 퐵퐶̅̅̅̅, therefore the value is 9푐푚, the reference to find the angles is the base angle, as a result, if ∡퐶퐴퐵 = 35, then ∡퐴퐶퐵 = 110.

Bibliography  Stewart, J., Redlin L. & Watson, S. (2012) Precálculo, Matemáticas para el cálculo, sexta edición. Gengage Learning. México.  Jiménez, D. (2005) Geometría el encanto de la forma. Colección Minerva, Los libros de la Nacional. Caracas-Venezuela.  Boyer, C. (2003). Historia de la Matemática. Editorial Alianza: Madrid.

Glossary  Triangle: A flat figure formed by three vertexes and three sides.  Straight Triangle: A triangle that has an internal right angle.  Isosceles Triangle: A triangle that is defined by having two sides of equal length.  Catheti: The sides opposing the acute angles of the right triangle.  Hypotenuse: The diagonal, or longest side, of the right triangle.  Pythagorean Theorem: The theorem that states that any right triangle presents the relation 푐2 = 푎2 + 푏2, c being the hypotenuse and a and b being the catheti.  Length: The distance between two points.  Straight angle: An angle that measures exactly 90°.